1970 AMC 12 Problem 9

Attempt Problem 9 of the 1970 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1970 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

Points

P

and

Q

are on line segment

AB,

and both points are on the same side of the midpoint of

AB.

Point

P

divides

AB

in the ratio

2:3,

and

Q

divides

AB

in the ratio

3:4.

PQ=2,

then the length of segment

AB

is:

12

28

70

75

105

Express

AP

and

AQ

as fractions of

AB

Use

AP=\frac{2AB}{5}

and

AQ=\frac{3AB}{7}

, then subtract

The division ratios give

AP=\frac25AB,\qquad AQ=\frac37AB.

Because the two points are on the same side of the midpoint,

\begin{aligned} PQ&=AQ-AP\\ &=\left(\frac37-\frac25\right)AB\\ &=\frac1{35}AB. \end{aligned}

Since

PQ=2,

we obtain

AB=70.

Therefore, the correct answer is C.

Problem 9 in other years